We study the energy efficiency of pinching-antenna systems (PASSs) by developing a consistent formulation for power distribution in these systems. The per-antenna power distribution in PASSs is not controlled explicitly by a power allocation policy, but rather implicitly through tuning of pinching couplings and locations. Both these factors are tunable: (i) pinching locations are tuned using movable elements, and (ii) couplings can be tuned by varying the effective coupling length of the pinching elements. While the former is feasible to be addressed dynamically in settings with low user mobility, the latter cannot be addressed at a high rate. We thus develop a class of hybrid dynamic-static algorithms, which maximize the energy efficiency by updating the system parameters at different rates. Our experimental results depict that dynamic tuning of pinching locations can significantly boost energy efficiency of PASSs.

Pinching-antenna systems (PASSs) are a recent flexible-antenna technology that is realized by attaching simple components, referred to aspinching elements, to dielectric waveguides. This work explores the potential of deploying PASS for uplink and downlink transmission in multiuser MIMO settings. For downlink PASS-aided communication, we formulate the optimal hybrid beamforming, in which the digital precoding matrix at the access point and the pinching locations on the waveguides are jointly optimized to maximize the achievable weighted sum-rate. We discuss the key challenges in this design problem and propose two low-complexity algorithms to iteratively update the precoding matrix and activated pinching locations.We further formulate the design problem for uplink transmission in a PASS and develop an iterative scheme for the underlying hybrid multiuser detection problem. We validate the proposed schemes through extensive numerical experiments. The results demonstrate that using a PASS, the throughput in both uplink and downlink is significantly enhanced compared to baseline MIMO architectures, such as massive MIMO and classical hybrid analog-digital designs. This highlights the great potential of the PASS, making it a promising reconfigurable antenna technology for next-generation wireless systems.

Quantization is a common approach to mitigate the communication cost of federated learning (FL). In practice, the quantized local parameters are further encoded via an entropy coding technique, such as Huffman coding, for efficient data compression. In this case, the exact communication overhead is determined by the bit rate of the encoded gradients. Recognizing this fact, this work deviates from the existing approaches in the literature and develops a novel quantized FL framework, called rate-constrained federated learning (RC-FED), in which we deploy the conventional entropy-constrained scalar quantization technique to quantize the gradients subject to both fidelity and data rate constraints. Particularly, we formulate this scheme, as a joint optimization in which the quantization distortion is minimized while the rate of encoded gradients is kept below a target threshold. This enables for a tunable trade-off between quantization distortion and communication cost. We analyze the convergence behavior of RC-FED, and show its superior performance against baseline quantized FL schemes on several datasets.

Quasi-Newton algorithms estimate the second-order information of loss landscape from its first-order information. They hence propose a promising solution for communication-efficient federated learning (FL), as they reduce the required number of training rounds while avoiding the necessity of exchanging local Hessians over the network. Despite that, the quasi-Newton approaches prove less effective in wireless FL, as the noisy aggregation in this case causes bias in the estimate of the Newton direction. This paper proposes a novel second-order wireless FL algorithm. The pivotal innovation lies in the server’s ability to estimate the global Hessian based on a window of noisy aggregations. The server acquires this ability by computing a stochastic estimator of the global Hessian under a Gaussian prior belief. Numerical experiments show that the proposed scheme can compute a less-biased estimator of the Newton direction, and hence a superior learning performance, as compared to the baseline.

This work invokes the notion of f-divergence to introduce a novel upper bound on the Bayes error rate of a general classification task. We show that the proposed bound can be computed by sampling from the output of a parameterized model. Using this practical interpretation, we introduce the Bayes optimal learning threshold (BOLT) loss whose minimization enforces a classification model to achieve the Bayes error rate. We validate the proposed loss for image and text classification tasks, considering MNIST, Fashion-MNIST, CIFAR10, and IMDb datasets. Numerical experiments demonstrate that models trained with BOLT achieve performance on par with or exceeding that of cross-entropy, particularly on challenging datasets. This highlights the potential of BOLT in improving generalization.

Differential privacy (DP) limits the impact of individual training data samples by bounding their gradient norms through clipping. Conventional clipping operations assign unequal scaling factors to sample gradients with different norms, leading to a direction mismatch between the true batch gradient and the aggregation of the clipped gradients. Applying a smaller but identical scaling factor to all sample gradients alleviates this direction mismatch; however, it intensifies the magnitude mismatch by excessively reducing the aggregation norm. This work proposes a novel clipping method, termed adaptive sigmoid (AdaSig), which uses a sigmoid function with an adjustable saturation slope to clip the sample gradients. The slope is adaptively adjusted during the training process to balance the trade-off between direction mismatch and magnitude mismatch, as the statistics of sample gradients evolve over the training iterations. Despite AdaSig’s adaptive nature, our convergence analysis demonstrates that differentially private stochastic gradient descent (DP-SGD) with AdaSig clipping retains the best-known convergence rate under non-convex loss functions. Evaluating AdaSig on sentence and image classification tasks across different datasets shows that it consistently improves learning performance compared with established clipping methods.

This work studies a low-complexity design for re-configurable intelligent surface (RIS)-aided multiuser multiple-input multiple-output systems. The base station (BS) applies receive antenna selection to connect a subset of its antennas to the available radio frequency chains. For this setting, the BS switching network, uplink precoders, and RIS phase-shifts are jointly designed, such that the uplink sum-rate is maximized. The principle design problem reduces to an NP-hard mixed-integer optimization. We hence invoke the weighted minimum mean squared error technique and the penalty dual decomposition method to develop a tractable iterative algorithm that approxi-mates the optimal design effectively. Our numerical investigations verify the efficiency of the proposed algorithm and its superior performance as compared with the benchmark.

Linear computation coding (LCC) has been developed in [1] as a new framework for the computation of linear functions. LCC significantly reduces the complexity of matrix-vector multiplication [1]. In basic LCC, storage is not restricted i.e. the wiring exponents are arbitrary integer exponents of 2.

Utilizing compressive sensing (CS), one can significantly reduce the number of required antenna elements in MIMO radar systems, while preserving a high spatial resolution. Most CS-based studies focus on individual processing of a single set of measurements collected from an stationary scene. In this paper, we propose a new scheme called multiple target measurements (MTM). This scheme uses the target movement to collect multiple sets of measurements from jointly sparse stationary scenes. Invoking approximate message passing, we develop a Bayesian-like iterative algorithm to recover the sparse scenes jointly. Our analytical and numerical investigations demonstrate that MTM can further reduce the array size required to achieve a desired spatial resolution.

It is widely believed that large IRS-aided MIMO settings maintain the fundamental features of massive MIMO systems. This work gives a rigorous proof that confirms this belief. We show that using a large passive IRS, the end-to-end MIMO channel between the transmitter and the receiver always hardens, even if the IRS elements are strongly correlated. For fading direct and reflection links between the transmitter and the receiver, our derivations demonstrate that for a large number of reflecting elements on the IRS, the capacity of the end-to-end channel is accurately approximated by a real-valued Gaussian random variable whose variance goes to zero as the number of IRS elements grows unboundedly large. The order of this drop depends on how the physical dimensions of the IRS grow. We derive this order explicitly. Numerical experiments show that the closed-form approximation very closely matches the histogram of the capacity term, even in practical scenarios. As a sample application of the results, we characterize the dimensional trade-off between the transmitter and the IRS. The result is intuitive: For a target performance, the larger the IRS is, the fewer transmit antennas are required.

Unlike the classical linear model, nonlinear generative models have been addressed sparsely in the literature of statistical learning. This work aims to shed light on these models and their secrecy potential. To this end, we invoke the replica method to derive the asymptotic normalized cross entropy in an inverse probability problem whose generative model is described by a Gaussian random field with a generic covariance function. Our derivations further demonstrate the asymptotic statistical decoupling of the Bayesian estimator and specify the decoupled setting for a given nonlinear model. The replica solution depicts that strictly nonlinear models establish an all-or-nothing phase transition: there exists a critical load at which the optimal Bayesian inference changes from perfect to an uncorrelated learning. Based on this finding, we design a new secure coding scheme which achieves the secrecy capacity of the wiretap channel. This interesting result implies that strictly nonlinear generative models are perfectly secured without any secure coding. We justify this latter statement through the analysis of an illustrative model for perfectly secure and reliable inference.

Inverse probability problems whose generative models are given by strictly nonlinear Gaussian random fields show the all-or-nothing behavior: There exists a critical rate at which Bayesian inference exhibits a phase transition. Below this rate, the optimal Bayesian estimator recovers the data perfectly, and above it the recovered data becomes uncorrelated. This study uses the replica method from the theory of spin glasses to show that this critical rate is the channel capacity. This interesting finding has a particular application to the problem of secure transmission: A strictly nonlinear Gaussian random field along with random binning can be used to securely encode a confidential message in a wiretap channel. Our large-system characterization demonstrates that this secure coding scheme asymptotically achieves the secrecy capacity of the Gaussian wiretap channel.

In intelligent reflecting surface (IRS)-aided multiple-input multiple-output (MIMO) systems, the IRS can be utilized to suppress the information leakage towards malicious terminals. This can lead to significant secrecy gains. This work exploits these gains via a tractablejointdesign of downlink beamformers and IRS phase-shifts. In this respect, we consider a generic IRS-aided MIMO wiretap setting and invoke fractional programming and alternating optimization to iteratively find the beamformers and phase-shifts that maximize the achievable weighted secrecy sum-rate. Our design is comprised of two low-complexity algorithms. Performance of the proposed algorithms are numerically evaluated and compared to the benchmark. The results reveal that integrating IRSs into MIMO systems not only boosts the secrecy performance, but also improves the robustness against passive eavesdropping.

This paper characterizes the performance of massive multiuser spatial modulation MIMO systems, when a regularized form of the least-squares method is used for detection. For a generic distortion function and right unitarily invariant channel matrices, the per-antenna transmit rate and the asymptotic distortion achieved by this class of detectors are derived. Invoking an asymptotic characterization, we address two particular applications. Namely, we derive the error rate achieved by the computationally-intractable optimal Bayesian detector, and we propose an efficient approach to tune LASSO-type detectors. We further validate our derivations through various numerical experiments.

Using intelligent reflecting surfaces (IRSs), wireless propagation channels can be manipulated such that information leakage to eavesdropping terminals in a multiple-input multiple-output (MIMO) setting is significantly suppressed. This observation illustrates the potential secrecy gains of IRS-aided MIMO systems. This work develops a novel low-complexity algorithm by which these potential gains are exploited. Invoking methods from fractional programming, the algorithm iteratively designs the digital precoder at the transmitter and tunes the IRS elements, such that the weighted secrecy sum-rate is maximized. It is shown that as the algorithm iterates, the weighted secrecy sum-rate evolves in a non-decreasing way. Numerical investigations confirm the efficiency of the proposed algorithm.

We introduce the new concept of computation coding. For linear functions, we present an algorithm to reduce the computational cost of multiplying an arbitrary given matrix with an unknown vector. It decomposes the given matrix into the product of codebook and wiring matrices whose entries are either zero or signed integer powers of two.For a typical implementation of deep neural networks, the proposed algorithm reduces the number of required addition units several times. To achieve the accuracy of 16-bit signed integer arithmetic for 4k-vectors, no multipliers and only 1.5 adders per matrix entry are needed.

Passively overheard massive multiple-input multiple-output (MIMO) settings are capable of suppressing eavesdroppers via narrow beamforming towards legitimate receivers. This implies that secrecy is obtained almost for free in these settings. This study shows that this is a valid property for a large class of low-complexity massive MIMO transmitters. The investigations consider two dominant approaches for complexity reduction, namely antenna selection and hybrid analog-digital precoding. It is shown that using either approach, the information leakage per achievable sum-rate vanishes as the number of transmit antennas grows large. The results demonstrate that, as the transmit array size grows large, the normalized information leakage obtained by antenna selection and hybrid analog-digital precoding converges to zero double-logarithmically and logarithmically, respectively. The analytical results are confirmed for various benchmark architectures via numerical simulations.

In this work, we propose a new single-RF MIMO architecture which enjoys high scalability and energy-efficiency. The transmitter in this proposal consists of a single RF illuminator radiating towards a reflecting surface. Each element on the reflecting surface re-transmits its received signal after applying a phase-shift, such that a desired beamforming pattern is obtained. For this architecture, the problem of beamforming is interpreted as linear regression and a solution is derived via the method of least-squares. Using this formulation, a fast iterative algorithm for tuning of the reflecting surface is developed. Numerical results demonstrate that the proposed architecture is fully compatible with current designs of reflecting surfaces.

In conventional Multiple-Input Multiple Output (MIMO) systems, each antenna requires its own Radio Frequency (RF) chain. Since each RF-chain includes several active components that have to be synchronised, costs and complexity become restrictive. The outphasing MIMO and the outphasing precoder architectures are possible approaches to mitigate this problem. The core of both architectures are Outphasing Elements (OEs), which are used to form a electronically controllable, passive antenna feed network. In this paper, these OEs are analysed with respect to component tolerances. The feasibility of the outphasing concept is demonstrated by the implementation of a hardware prototype OE. The performance of this prototype is measured and compared to theoretical predictions with good agreement.

Most detection algorithms in spatial modulation (SM) are formulated as linear regression via the regularized least-squares (RLS) method. In this method, the transmit signal is estimated by minimizing the residual sum of squares penalized with some regularization. This paper studies the asymptotic performance of a generic RLS-based detection algorithm employed for recovery of SM signals. We derive analytically the asymptotic average mean squared error and the error rate for the class of bi-unitarily invariant channel matrices. The analytic results are employed to study the performance of SM detection via the box-LASSO. The analysis demonstrates that the performance characterization for i.i.d. Gaussian channel matrices is valid for matrices with non-Gaussian entries, as well. This justifies the partially approved conjecture given in [1]. The derivations further extend the former studies to scenarios with non-i.i.d. channel matrices. Numerical investigations validate the analysis, even for practical system dimensions.

Joint user selection and precoding in multiuser MIMO settings can be interpreted as group sparse recovery in linear models. In this problem, a signal with group sparsity is to be reconstructed from an underdetermined system of equations. This paper utilizes this equivalent interpretation and develops a computationally tractable algorithm based on the method of group LASSO. Compared to the state of the art, the proposed scheme shows performance enhancements in two different respects: higher achievable sum-rate and lower interference at the non-selected user terminals.

The large-system performance of maximum-a-poste-rior estimation is studied considering a general distortion function when the observation vector is received through a linear system with additive white Gaussian noise. The analysis considers the system matrix to be chosen from the large class of rotationally invariant random matrices. We take a statistical mechanical approach by introducing a spin glass corresponding to the estimator, and employing the replica method for the large-system analysis. In contrast to earlier replica based studies, our analysis evaluates the general replica ansatz of the corresponding spin glass and determines the asymptotic distortion of the estimator for any structure of the replica correlation matrix. Consequently, the replica symmetric as well as the replica symmetry breaking ansatz with$b$steps of breaking is deduced from the given general replica ansatz. The generality of our distortion function lets us derive a more general form of the maximum-a-posterior decoupling principle. Based on the general replica ansatz, we show that for any structure of the replica correlation matrix, the vector-valued system decouples into a bank of equivalent decoupled scalar systems followed by maximum-a-posterior estimators. The structure of the decoupled system is further studied under both the replica symmetry and the replica symmetry breaking assumptions. For$b$steps of symmetry breaking, the decoupled system is found to be an additive system with anon-Gaussiannoise term given as the sum of an independent Gaussian random variable with$b$non-Gaussian impairment terms which depend on the input symbol. The general decoupling property of the maximum-a-posterior estimator leads to the idea of a replica simulator which represents the replica ansatz through the state evolution of a transition system described by its corresponding decoupled system. As an application of our study, we investigate large compressive sensing systems by considering the$\ell _{p}$norm minimization recovery schemes. Our numerical investigations show that the replica symmetric ansatz for$\ell _{0}$norm recovery fails to give an accurate approximation of the mean square error as the compression rate grows, and therefore, the replica symmetry breaking ansätze are needed in order to assess the performance precisely.

This paper proposes the class of generalized least-square-error (GLSE) precoders for multiuser massive multiple-input multiple-output (MIMO) systems. For a generic transmit constellation, the GLSE precoders minimize the interference at user terminals assuring that some given constraints on the transmit signals are satisfied. The general form of these precoders enables us to impose multiple restrictions at the transmit signal, such as limited peak power and restricted number of active transmit antennas. The performance of these precoders is analyzed in the large-system limit. It is shown that the output symbols are identically distributed, and their statistics are described with an equivalent scalar GLSE precoder. To demonstrate the applications of the proposed framework, we employ the GLSE precoding to form transmit signals over a discrete alphabet and to select an effective subset of transmit antennas. Our investigations show that a computationally efficient GLSE precoder requires 41% less active transmit antennas than the conventional selection protocols in order to achieve a given level of input-output distortion.

In massive MIMO wiretap settings, the base station can significantly suppress eavesdroppers by narrow beamforming toward legitimate terminals. Numerical investigations show that by this approach, secrecy is obtained at no significant cost. We call this property of massive MIMO systems "secrecy for free" and show that it not only holds when all the transmit antennas at the base station are employed, but also when only a single antenna is set active. Using linear precoding, the information leakage to the eavesdroppers can be sufficiently diminished, when the total number of available transmit antennas at the base station grows large, even when only a fixed number of them are selected. This result indicates that passive eavesdropping has no significant impact on massive MIMO systems, regardless of the number of active transmit antennas.

This paper studies the asymptotic performance of maximum-a-posteriori estimation in the presence of prior information. The problem arises in several applications such as recovery of signals with non-uniform sparsity pattern from underdetermined measurements. With prior information, the maximum-a-posteriori estimator might have asymmetric penalty. We consider a generic form of this estimator and study its performance via the replica method. Our analyses demonstrate an asymmetric form of the decoupling property in the large-system limit. Employing our results, we further investigate the performance of weighted zero-norm minimization for recovery of a non-uniform sparse signal. Our investigations illustrate that for a given distortion, the minimum number of required measurements can be significantly reduced by choosing weighting coefficients optimally.

The typical approach for recovery of spatially correlated signals is regularized least squares with a coupled regularization term. In the Bayesian framework, this algorithm is seen as a maximum-a-posterior estimator whose postulated prior is proportional to the regularization term. In this paper, we study distributed sensing networks in which a set of spatially correlated signals are measured individually at separate terminals, but recovered jointly via a generic maximum-a-posterior estimator. Using the replica method, it is shown that the setting exhibits the decoupling property. For the case with jointly sparse signals, we invoke Bayesian inference and propose the “multi-dimensional soft thresholding” algorithm which is posed as a linear programming. Our investigations depict that the proposed algorithm outperforms the conventional l2,1-norm regularized least squares scheme while enjoying a feasible computational complexity.

In this paper, we study the impacts of transmit antenna selection on the secrecy performance of massive MIMO systems. We consider a wiretap setting in which a fixed number of transmit antennas are selected and then confidential messages are transmitted over them to a multi-antenna legitimate receiver while being overheard by a multi-antenna eavesdropper. For this setup, we derive an accurate approximation of the instantaneous secrecy rate. Using this approximation, it is shown that in some wiretap settings under antenna selection the growth in the number of active antennas enhances the secrecy performance of the system up to some optimal number and degrades it when this optimal number is surpassed. This observation demonstrates that antenna selection in some massive MIMO settings not only reduces the RF-complexity, but also enhances the secrecy performance. We then consider various scenarios and derive the optimal number of active antennas analytically using our large-system approximation. Numerical investigations show an accurate match between simulations and the analytic results.

This paper proposes nonlinear least square error (LSE) precoders for multiuser MIMO broadcast channels. The LSE precoders are designed such that the discrete output signals are from a predefined set. This predefined set allows us to model several signal constraints such as peak power constraint, constant envelope, and discrete constellations. We study the large-system performance of these precoders via the replica method from statistical physics, and derive a closed-form expression for the asymptotic distortion. Our results demonstrate that an LSE precoder with the output peak-to-average power ratio of 3 dB can perform similar to the regularized zero forcing (RZF) precoder. As the peak-to-average power ratio reduces to one, the constant envelope precoder is recovered. The investigations show that the performance of the RZF precoder is achieved by a constant envelope precoder with 20% additional transmit antennas. For M-phase shift keying constellations, our analysis gives a lower bound on the asymptotic distortion which is tight for moderate antenna-to-user ratios and deviates as the ratio grows. We improve this bound by deriving the replica solution under one-step of replica symmetry breaking. Our numerical investigations for this case show that the bound is tight for antenna-to-user ratios less than 5.

This paper studies the impact of transmit antenna selection on the secrecy performance of massive MIMO wiretap channels. We consider a scenario in which a multi-antenna transmitter selects a subset of transmit antennas with the strongest channel gains. Confidential messages are then transmitted to a multi-antenna legitimate receiver while the channel is being overheard by a multi-antenna eavesdropper. For this setup, we approximate the distribution of the instantaneous secrecy rate in the large-system limit. The approximation enables us to investigate the optimal number of selected antennas which maximizes the asymptotic secrecy throughput of the system. We show that increasing the number of selected antennas enhances the secrecy performance of the system up to some optimal value, and that further growth in the number of selected antennas has a destructive effect. Using the large-system approximation, we obtain the optimal number of selected antennas analytically for various scenarios. Our numerical investigations show an accurate match between simulations and the analytic results even for not so large dimensions.

Consider a fading Gaussian MIMO channel with Nttransmit and Nrreceive antennas. The transmitter selects Ltantennas corresponding to the strongest channels. For this setup, we study the distribution of the input-output mutual information when Ntgrows large. We show that, for any Nrand Lt, the distribution of the input-output mutual information is accurately approximated by a Gaussian distribution whose mean grows large and whose variance converges to zero. Our analysis depicts that, in the large limit, the gap between the expectation of the mutual information and its corresponding upper bound, derived by applying Jensen's inequality, converges to a constant which only depends on Nrand Lt. The result extends the scope of channel hardening to the general case of antenna selection with multiple receive and selected transmit antennas. Although the analyses are given for the large-system limit, our numerical investigations indicate the robustness of the approximated distribution even when the number of antennas is not large.

A general class of nonlinear Least Square Error (LSE) precoders in multi-user multiple-input multiple-output systems is analyzed using the replica method from statistical mechanics. A single cell downlink channel with N transmit antennas at the base station and K single-antenna users is considered. The data symbols are assumed to be iid Gaussian and the precoded symbols on each transmit antenna are restricted to be chosen from a predefined set X. The set X encloses several well-known constraints in wireless communications including signals with peak power, constant envelope signals and finite constellations such as Phase Shift Keying (PSK). We determine the asymptotic distortion of the LSE precoder under both the Replica Symmetry (RS) and the one step Replica Symmetry Breaking (1-RSB) assumptions. For the case of peak power constraint on each transmit antenna, our analyses under the RS assumption show that the LSE precoder can reduce the peak to average power ratio to 3dB without any significant performance loss. For PSK constellations, as N/K grows, the RS assumption fails to predict the performance accurately and therefore, investigations under the 1-RSB assumption are further considered. The results show that the 1-RSB assumption is more accurate.

This paper studies the large-system performance of Least Square Error (LSE) precoders which minimize the input-output distortion over an arbitrary support subject to a general penalty function. The asymptotics are determined via the replica method in a general form which encloses the Replica Symmetric (RS) and Replica Symmetry Breaking (RSB) ansätze. As a result, the “marginal decoupling property” of LSE precoders for b-steps of RSB is derived. The generality of the studied setup enables us to address special cases in which the number of active transmit antennas are constrained. Our numerical investigations depict that the computationally efficient forms of LSE precoders based on “li-norm” minimization perform close to the cases with “zero-norm” penalty function which have a considerable improvements compared to the random antenna selection. For the case with BPSK signals and restricted number of active antennas, the results show that RS fails to predict the performance while the RSB ansatz is consistent with theoretical bounds.

The large-system decoupling property of a MAP estimator is studied when it estimates the i.i.d. vector x from the observation y = Ax + z with A being chosen from a wide range of matrix ensembles, and the noise vector z being i.i.d. and Gaussian. Using the replica method, we show that the marginal joint distribution of any two corresponding input and output symbols converges to a deterministic distribution which describes the input-output distribution of a single user system followed by a MAP estimator. Under the bRSB assumption, the single user system is a scalar channel with additive noise where the noise term is given by the sum of an independent Gaussian random variable and b correlated interference terms. As the bRSB assumption reduces to RS, the interference terms vanish which results in the formerly studied RS decoupling principle.

In this paper, we investigate the problem of the empirical coordination in a triangular multiterminal network. A triangular multiterminal network consists of three terminals where two terminals observe two external i.i.d correlated sequences. The third terminal wishes to generate a sequence with desired empirical joint distribution. For this problem, we derive inner and outer bounds on the empirical coordination capacity region. It is shown that the capacity region of the degraded source network and the inner and outer bounds on the capacity region of the cascade multiterminal network can be directly obtained from our inner and outer bounds. For a cipher system, we establish key distribution over a network with a reliable terminal, using the results of the empirical coordination. As another example, the problem of rate distortion in the triangular multiterminal network is discussed in which a distributed doubly symmetric binary source is available.